3 Two-Dimensional Linear Algebra
3 Two-Dimensional Linear Algebra 3.1 Basic Rules Vectors are the way we represent points in 2, 3 and 4, 5, 6, ..., n, ...-dimensional space. There even exist spaces of vectors which are said to be infinite- dimensional! A vector can be considered as an object which has a length and a direction, or it can be thought of as a point in space, with coordinates representing that point. In terms of coordinates, we shall use the notation µ ¶ x x = y for a column vector. This vector denotes the point in the xy-plane which is x units in the horizontal direction and y units in the vertical direction from the origin. We shall write xT = (x, y) for the row vector which corresponds to x, and the superscript T is read transpose. This is the operation which flips between two equivalent represen- tations of a vector, but since they represent the same point, we can sometimes be quite cavalier and not worry too much about whether we have a vector or its transpose. To avoid the problem of running out of letters when using vectors we shall also write vectors in the form (x1, x2) or (x1, x2, x3, .., xn). If we write R for real, one-dimensional numbers, then we shall also write R2 to denote the two-dimensional space of all vectors of the form (x, y). We shall use this notation too, and R3 then means three-dimensional space. By analogy, n-dimensional space is the set of all n-tuples {(x1, x2, ..., xn): x1, x2, ..., xn ∈ R} , where R is the one-dimensional number line.
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